Download Neutral Weak Interactions

Neutral Weak Interactions
• The Z 0 Boson:
• Feynman Rules
• The Weak Mixing Angle
• Resonance in e+ e− Scattering
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Fermi’s Theory
• Recall that Fermi modeled the weak interaction as a four fermion contact
interaction with a coupling of GF :
f3
f4
f1
f2
˜ˆ
˜
GF ˆ ¯ µ
5
5
¯
M = √ f3 γ (1 − γ )f1 f4 γµ (1 − γ )f2 ∼ GF E 2
2
• From quantum scattering theory, one has a problem if |M| ∼ 1
This is known as the unitarity bound.
• For the Fermi model, we can state that it will not work beyond E ∼ 300 GeV.
This is resolved by incorporating a W -boson (MW ∼ 80 GeV) into the theory
of weak interactions
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Unitarity Bounds I
• Even after we incorporate W ± bosons into a theory of the weak interaction,
another unitarity bound is encountered
• The amplitude for the e+ e− → W + W − process is a problem assuming that
it proceeds by the Feynman diagram:
W−
W+
νe
e−
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e+
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Restoring Unitarity
• In order to make the weak interaction self-consistent, we require two
additional contributions to the e+ e− → W + W − scattering process:
W−
W+
W−
γ
e−
W+
Z0
e+
e−
e+
• We require two neutral bosons, the γ and the Z 0 to fix the unitarity problem
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Discovery of the Z 0
• The Z 0 was indirectly observed in 1973 in the Gargamelle experiment at CERN
via the processes
ν̄µ + e
→
ν̄µ + e
ν̄µ + N
→
ν̄µ + X
• The forward-backward asymmetry (AF B ) in e+ e− → µ+ µ− also indirectly
showed the need for the Z 0
• The Z 0 was directly observed in 1983 by the UA1 and UA2 detectors at CERN
via proton-antiproton collisions.
• The Z 0 is slightly heavier than the W ± , with
MZ = 91.1876(21) GeV
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The Weak Mixing Angle
• Many of the parameters of the electroweak interaction are related to each other.
The masses and couplings are related by weak mixing or Weinberg angle (θw )
• For example, the masses of the W and Z bosons are related by
MW = MZ cos θw
• The vertex factor (gZ ) for the Z 0 is related to the W vertex factor (gW )
gz = gw cos θw
• Both gw and gz are related to the QED coupling constant ge
gw =
ge
sin θw
gz =
ge
sin θw cos θw
This is why the weak force is inherently stronger than the electromagnetic
force.
• Experimentally,
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sin2 θw (MZ ) = 0.23120(15)
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Feynman Rules for the Z 0
• The Z 0 propagator looks just like that of the W :
„
«
qµ qν
−i gµν − M 2
Z
2
q 2 − MZ
• The Z 0 bosons mediate neutral current (NC) weak interactions.
They couple to fermions via
f
−igz
2
γ µ (cV − cA γ 5 )
Z0
f
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Fermion Couplings to the Z 0
• The vector and axial couplings cV and cA are specified by the GWS model:
f
cV
cA
νℓ
+ 21
+ 12
ℓ−
− 21 + 2 sin2 θw
− 12
qu
+ 12 −
4
3
sin2 θw
+ 12
qd
− 12 +
2
3
sin2 θw
− 12
• Note that the Z 0 does not change the lepton or quark flavor. The Standard
Model has no flavor-changing neutral currents (FCNC) at tree level.
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Gauge Boson Self-Couplings
• Just like QCD, the electroweak bosons can interact with each other:
W+
W−
X
W+
Y
Z0, γ
W−
where (X, Y ) can be (γ, γ), (γ, Z 0 ), (Z 0 , Z 0 ), or (W + , W − ).
Consult Appendix D of Griffiths for vertex factors.
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Compare γ and Z 0 diagrams
• The Z 0 couples to every charged fermion, just like the photon does.
γ→ff
Z→ff
• The similarity between the γ and the Z 0 made it difficult to detect the Z 0
2 )−1
because at low energies. The QED effects dominate due to the (q 2 − MZ
factor in the Z propogator.
• For example, the forward-backward asymmetry became more pronounced as
the centre-of-mass energy became larger
• Unlike the photon, the Z 0 also couples to neutrinos.
Z→νν
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Example: e+ e− → f f¯
• We now have a new diagram for the e+ e− → f f¯ process
4 : f¯
3:f
Z0
1 : e−
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2 : e+
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The Scattering Amplitude
• The amplitude is
M
=
2
„
qµ qν
2
MZ
« – −i gµν −
» „
6
−igz µ f
γ (cV − cfA γ 5 ) v3 6
i ū4
4
2
2
q 2 − MZ
« –
» „
−igz ν e
γ (cV − ceA γ 5 ) u1
× v̄2
2
«3
7
7
5
2 , the Z 0 -mediated diagram is similar in structure to
• At low energies, q 2 ≪ MZ
−2
the QED diagram. The γ amplitude would dominate as q −2 ≫ MZ
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Scattering amplitude at q 2 = MZ2 : I
• If q 2 is not small, we can no longer simplify the Z 0 -propagator.
Keeping the full propagator,
M
=
−
gz2
i
h
f 5
µ f
ū4 γ (cV − cA γ )v3
2)
4(q 2 − MZ
˜
ˆ
× v̄2 γ ν (ceV − ceA γ 5 )u1
• If we can neglect all fermion masses, the
qµ qν
2
MZ
gµν −
qµ qν
2
MZ
!
part of the propagator will
contribute nothing, since we can write q as either p1 + p2 or p3 + p4 .
• We can write q as either p1 + p2 or p3 + p4
then the /
q factors lead to combinations like ū4 /
p4 and /
p3 v3 ,
which, by the Dirac equation, are ū4 m4 and −m3 v3 .
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Scattering amplitude at q 2 = MZ2 : II
• The scattering amplitude can be written as
M
=
E
D
2
|M|
=
i
h
gz2
f 5
µ f
ū4 γ (cV − cA γ )v3
−
2)
4(q 2 − MZ
˜
ˆ
× v̄2 γµ (ceV − ceA γ 5 )u1
"
#2
i
h
2
gz
f 5
f 5
ν f
µ f
p4
p3 γ (cV − cA γ )/
Tr γ (cV − cA γ )/
2)
8(q 2 − MZ
˜
ˆ
e 5
e
e 5
e
p2
p1 γν (cV − cA γ )/
× Tr γµ (cV − cA γ )/
• The traces are best evaluated by first bringing the cV and cA terms together:
(cV − cA γ 5 )/
p3 γ ν (cV − cA γ 5 )
=
=
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(cV − cA γ 5 )2 /
p3 γ ν
p3 γ ν − 2cV cA γ 5 /
p3 γ ν
(c2V + c2A )/
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Scattering amplitude at q 2 ≈ MZ2 : III
• The cross section for Z 0 -mediated e+ e− → f f¯ is
!2
2
gz E
1
f 2
f 2
e 2
e 2
)
+
(c
[(c
)
+
(c
)
][(c
σ =
A) ]
V
V
A
2]
3π 4[(2E)2 − MZ
• Note that this cross section blows up when E = MZ /2.
• This is much more serious than the infinite cross section for Rutherford
scattering because this (Z 0 ) divergence can be traced all the way back to the
amplitude.
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Propogator for unstable particles
• The source of the problem is that the kinematics are such that e+ e− → Z 0 is a
physically allowable process even without a subsequent decay to f f¯.
• As a result, we need to modify the Z 0 -propagator in order to account for the
instability of the Z 0 . Here’s what we do:
1. We recall the familiar configuration-space wavefunction of a stable
particle:
Ψ(r, t) = ψ(r)e−iEt
2. Since the particle is stable, the probability of finding the particle
somewhere is always equal to 1 since the wavefunction i s normalized:
Z
P (t) =
|Ψ|2 d3 r = 1
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3. If the particle is unstable, we expect the probability of finding the particle to
fall off with time according to the decay rate Γ
Z
P (t) =
|Ψ|2 d3 r = e−Γt
4. In the particle rest frame, this means that
Ψ(r, t) = ψ(r)e−iM t−
Γt
2
5. We then apply the substitution M → M − iΓ
to the propagator of an unstable
2
particle and assume that Γ is sufficiently small that we can neglect the Γ2 term:
1
q2 − M 2
→
≃
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1
q 2 − (M − iΓ/2)2
1
q 2 − M 2 + iM Γ
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Cross Section
• With the modification to the Z 0 propagator,
1
2
q 2 − MZ
→
1
2 + iM Γ
q 2 − MZ
Z Z
• The cross section takes the form
σ
∼
1
2 ]2 + (M Γ )2
[(2E)2 − MZ
Z Z
• This is known as a Breit-Wigner resonance.
The height and width of the resonance peak are determined by the decay
width ΓZ .
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Measurement of the Z 0 Peak : I
The Breit-Wigner shape is modified by higher-order (radiative) effects
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Measurement of the Z 0 Peak : II
e+ e− → f f¯
QED dominates at low energies
„
«
E 4
σZ
≃2
σγ
MZ
The Z 0 -mediated process which dominates near the resonance peak
«
„
σZ
1 MZ 2
≃ 200
≃
σγ
8 ΓZ
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Number of light neutrino generations
Z 0 → νν is allowed
Each ν species contributes to the
total width ΓZ
Since σ ∝ ΓZ the cross section
will depend on the number of
(light) neutrino generations
The data shows that there cannot be a 4th lepton generation with a light neutrino.
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The Z 0 Peak at CERN
• Precise measurements of electroweak parameters (MW , MZ , and sin2 θw ) also
shed light on other Standard Model parameters such as mt and mH .
• In the early days at LEP (started in 1989), a number of unusual systematic
effects needed to be accounted for in order to measure these parameters
accurately:
1. Tidal distortions of the ring
2. Water levels in nearby Lake Geneva
3. Correlations with the TGV
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Water levels in nearby Lake Geneva
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Tidal distortions of the ring
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Correlations with the TGV
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LEP-II WW and ZZ results:
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LEP-II WW and ZZ results:
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LEP-II WW and ZZ results:
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ATLAS W and Z results: W − → e− νe
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ATLAS W and Z results W − → τ − ντ
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ATLAS W and Z results Z 0 → e+ e−
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ATLAS W and Z results Z 0 → τ + τ −
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ATLAS W and Z results
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ATLAS W and Z results
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